Electronic spectra of one-dimensional nano-quasi-periodic systems under bias

We investigate the properties of the energy spectra of quantum one-dimensional nano-quasi-crystals in the presence of external electric fields. These systems are modelled by means of finite sequences, ordered according to a Fibonacci rule which constituted of two blocks A$A$ (constant potentials of different heights defined on finite intervals) and B$B$ (delta potentials of different intensities). We use the electric field ability of producing Stark ladders in periodic systems to obtain well separated energy levels and to study the evolution of these levels when disorder is introduced. We show that this effect also allows us to predict the approximate position of the levels in the disordered system, in spite of its chaotic appearance at first view. We show, against the usual belief, that the n$n$th Stark ladder in general is not formed exclusively from the levels of the n$n$th band. The disorder is introduced in two different ways: by changing the distribution of the blocks or by changing the values of the delta potential intensities. In both cases we start from electrified periodic structures which are gradually perturbed to obtain electrified quasi-periodic structures. We show that the use of Fibonacci sequences as a particular case is not crucial and one can use the electric field to analyze any other type of quasi-periodic systems.